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Properties

Label 1-211-211.107-r0-0-0
Degree $1$
Conductor $211$
Sign $0.886 - 0.461i$
Analytic cond. $0.979879$
Root an. cond. $0.979879$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)5-s + (0.309 − 0.951i)6-s + (0.309 − 0.951i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.809 − 0.587i)10-s + (−0.809 + 0.587i)11-s + 12-s + (0.309 − 0.951i)13-s + 14-s + 15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)5-s + (0.309 − 0.951i)6-s + (0.309 − 0.951i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.809 − 0.587i)10-s + (−0.809 + 0.587i)11-s + 12-s + (0.309 − 0.951i)13-s + 14-s + 15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $0.886 - 0.461i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ 0.886 - 0.461i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6203551073 - 0.1518517538i\)
\(L(\frac12)\) \(\approx\) \(0.6203551073 - 0.1518517538i\)
\(L(1)\) \(\approx\) \(0.7131985590 + 0.1206495772i\)
\(L(1)\) \(\approx\) \(0.7131985590 + 0.1206495772i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.20109598508020821494117868443, −26.23993432171960465127145198436, −24.34356246921746382345270833464, −23.76336424079878628771651785759, −22.9100802081226676953038273430, −21.89406139048772245369856947540, −21.10474442350882773977353871853, −20.56899623062486849412430241817, −18.97403502698956640330494846965, −18.62371343750453902836786923728, −17.24220174988157556267534420247, −16.124742110665156603974744595031, −15.32150837931522134951879549458, −14.19626699546792764491497907831, −12.59358652639851326727133226286, −12.15028424479726686077099258772, −11.2071142042530389917387204610, −10.395731250838364572386475957071, −9.05455370791051116630712895633, −8.291200855693344924830232379544, −6.142184282991539805285850562152, −5.18091959230893925204969304175, −4.317635871802084565434411051623, −3.17626747499790011300814063108, −1.36604020896095494845972677483, 0.54145211680029680826015204526, 3.02997284390082207356278742978, 4.4884740651788747746792476272, 5.35585587411784532641601324898, 6.759295511033992871073051693906, 7.46821878868871459582832999107, 8.03506497494389619608443452420, 10.009272998429504795357703654592, 11.09399195606361987727646830480, 12.10348413900760121856333351702, 13.246588652894384019412258314780, 13.936598702473743123707454720791, 15.407343715374783882782787629951, 15.87336609925608130347457198197, 17.2004493902271924849724187050, 17.82626171749553111515008398452, 18.63992396451888682661003398716, 19.874959454984781552745001415719, 21.191848108888197525469311057645, 22.62498608757127605778911202965, 22.9979131728687496666387503222, 23.62315728537568052494853410965, 24.51529168374158304054344300915, 25.59119310400204641799523576045, 26.600149642740637359175955415198

Graph of the $Z$-function along the critical line